Paulions

A 2-1. three-necked flask is thoroughly dried and fitted with a large dry-ice condenser, a mechanical stirrer, a nitrogen inlet, and a powder funnel in an efficient hood. With nitrogen flowing through the system, 62.5 g. (1.60 moles) of commercial sodium amide (Note 1) is added rapidly. (Paulion Sodium amide is corrosive and readily decomposes in the presence of moisture.) The funnel is replaced by a gas-inlet tube, the condenser is filled with a mixture of dry ice and acetone, and ca. 400 ml. of liquid... 

Energy operator for a molecular crystal with fixed molecules in the second-quantization representation. Paulions and Bosons... 

In our case the perturbation is represented by the nonresonant terms in the paulion Hamiltonian... 

First let us remark that the replacement of Pauli operators by Bose operators applied in this chapter is only approximate since the occupation numbers of paulions can be either 0 or 1, whereas the occupation numbers for bosons can take all nonnegative integer values 0,1,2,.., .28 Therefore the replacement of the operators Ps and Pj by Bose operators can provoke uncontrolled errors in all... 

We can, however, improve the transition from Pauli operators to Bose operators Bs and B] requiring that for an arbitrary number of bosons the number of paulions will equal 0 or 1 (90). To this aim we write the Pauli operators in the form... 

The occupation number operator for paulions Ls = P) Ps is thereby expressed by the occupation number operator for bosons by the relation... 

We can easily verify that Ls = 0 corresponds to states with an even number of bosons, whereas Ls = 1 corresponds to states with an odd number of bosons. Thus the transformations (3.192) and (3.193) do not create states with unphys-ical number of paulions (i.e. numbers Ls > 1). 

The problem of the kinematic interaction between two paulions is similar to the problem of localized states of an exciton in the presence of a vacancy (98). Indeed, the kinematic interaction governing the relative motion of two Pauli particles is formally analogous to the one-particle potential created by a vacancy, which cannot be occupied by an exciton. In this case the equation determining the localized exciton state energy E is... 

The strongly localized nature of low-energy polariton states should affect many processes such as light scattering and nonlinear phenomena as well as temperature-induced diffusion of polaritons. Manifestations of the localized polariton statistics (Frenkel excitons are paulions exhibiting properties intermediate between Fermi and Bose particles) in the problem of condensation also appear interesting and important. 

In this section, using the results of Section 3.11, we consider collective properties of an ideal gas of paulions, i.e. of a system described by the Hamiltonian (15.5) corresponding to the case of a vanishing operator of the dynamic interaction between elementary excitations, i.e. with i7int = 0.59... 

The states of bound bosons are not unphysical, because by transitions to these states the number of paulions, which is not equal to the number of bosons at the lattice point (see... 

Making use of the scattering length above obtained and recalling the results of (12), (13), we find that if k = 0 corresponds to the minimum of the excitonic band, the spectrum of an ideal paulion gas is given by... 

For concentrations No < 1018 1/cm3 and a 5 10 8 cm the wavevector j4iraNo < 105 cm-1, i.e. is of the order of the light wavevector. The total energy shift of states with k C y/Ana No appearing because excitons are paulions and not bosons for the same parameters equals... 

The conclusion on the possibility of condensation of elementary excitations in momentum space with Hamiltonian (15.5) coincides with that made in the article by Bocchieri and Seneci (14) where the problem of condensation in a crystal lattice of an ideal paulion gas has been considered without the transformation from paulions to bosons. For this reason the spectrum of the elementary excitations of the condensate has not been obtained. 

The generalization of the representation of paulion operators in terms of bosonic ones for the case of truncated oscillators of higher ranks is derived in the paper (15). The authors of this paper used this generalization to introduce a new constraint-free bosonic description of truncated oscillator systems. This result can be important in consideration of collective properties of Frenkel excitons in organic crystals with account of multilevel molecular structures and mixing of molecular configurations. 

In the opposite case of narrow exciton bands, when the interaction (15.15) leads to sticking together of excitons into pairs, triplets, and more numerous excitonic droplets , a more appropriate description is that in terms of excitons-paulions. The process of exciton coagulation can be described with the well-known methods of colloid statistics (21). We notice here that the size distribution of exciton droplets will depend on the exciton lifetime, bimolecular quenching... 

The exciton-exciton and polariton-polariton kinematic interactions in a crystalline organic monolayer and in an organic microcavity have been considered in (26). The kinematic interactions in this paper are derived using for Frenkel excitons the transformation from paulions to bosons (see Ch. 3). 

Another situation can be observed in one-dimensional crystals. Chesnut and Suna (30) have shown that if in a one-dimensional paulion gas only interactions with nearest-neighbors are taken into account, the energy operator for this case can be represented as the energy operator of an ideal gas of fermions. 

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